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On the measurement of strain rate in an oscillatory baffled column using particle image velocimetry
Chemical Engineering Science 55 (2000) 3195}3208
On the measurement of strain rate in an oscillatory ba%ed column using particle image velocimetry X. Ni *, J. A. Cosgrove , A. D. Arnott, C. A. Greated, R. H. Cumming Department of Mechanical and Chemical Engineering, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, UK Department of Physics and Astronomy, The University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK Applied Sciences Section, The University of Teesside, Middlesborough, TS1 3BA, UK Received 21 January 1999; accepted 13 September 1999
Abstract We report, for the "rst time, our direct experimental measurement of velocity vectors and strain-rate distributions in an oscillatory ba%ed column (OBC) using time-resolved particle image velocimetry (PIV). The technique allowed a time series of spatial velocity maps to be obtained for several phases per oscillation cycle and the results, comprising several thousand such maps, illustrate in detail the variations of velocity and strain rate in a ba%ed region over a range of oscillation amplitudes and oscillation frequencies and provide a deep insight into the mixing, vortex convection and transport mechanism in such a device. The results show that an OBC not only provides enhanced mixing, but also o!ers low strain rates, which are lower than those in stirred tank vessels. We have also reported for the "rst time the quantitative correlation between the strain rate and the power dissipation in an OBC. Using the PIV technique we are able to quantify the strain rates experienced in an OBC. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Oscillatory ba%ed column; Particle image velocimetry; Velocity vector; Strain rate; Vortices; Oscillation frequency; Oscillation amplitude
1. Introduction Many #ows present signi"cant challenges to experimental measurement, analysis and modelling, especially those where the motion is very complex in both space and time domains. A typical example is oscillatory (unsteady) #ow in a ba%ed tube. In such a device, #uid mixing is achieved by eddies that are generated when #uid passes through a set of equally spaced stationary ori"ce ba%es. Those periodically formed vortices can be controlled by a combination of geometrical and operational parameters, such as, ba%e diameter, ba%e spacing, oscillation frequency and oscillation amplitude. Under certain operational conditions, an oscillatory ba%ed column (OBC) can be operated as either a plug-#ow reactor or an enhanced mixing device (Brunold, Hunns, Mackley & Thompson, 1989; Dickens, Mackley & Williams, 1989; Mackley & Ni, 1991,1993). For a given ba%e geometry, the #uid mechanical condition in an OBC is controlled by the oscillatory Reynolds number, Re , and the * Corresponding author. Tel.: #0131-451-3781x4723; fax: #0131451-3077. E-mail address: [email protected] (X. Ni)
Strouhal number, St, de"ned as Dx u Re " , l
(1)
D St" , (2) 4px where D is the column diameter (m), x the oscillation amplitude (m), u the angular frequency of oscillation ("2pf ), f is the oscillation frequency (Hz), and l the kinematic viscosity of #uid (m/s). There are essentially two models for estimating the power input in an OBC: the quasi-steady #ow model and the eddy acoustic model (Baird & Stonestreet, 1995). The former assumes that oscillatory #ows are &quasi-steady', i.e. the instantaneous pressure drop in a time-periodic #ow is assumed to be identical to the pressure drop that would be obtained at a steady velocity of the same magnitude of the instantaneous velocity. The power input for this model is valid for higher oscillation amplitudes and lower frequencies, e.g. 5}30 mm, 0.5}2 Hz. For the same scale of the column and the same viscosity of #uid as used in this work, this corresponds to the range of the oscillatory Reynolds number from 785 to 18 850 and the range of the Strouhal number from 0.8 to 0.13. The
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 7 7 - 1
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Nomenclature B C " D D k n N N P/< Re St u v x x y y Greek letters a c e l o p u
constant ori"ce discharge coe$cient ("0.7) column diameter, m ori"ce diameter, m constant index of power-law #uid number of ba%es per unit length, /m speed of stirrer, r.p.s. power density, W/m oscillatory Reynolds number de"ned by Eq. (1) Strouhal number de"ned by Eq. (2) radial velocity component, m/s axial velocity component, m/s axial axis amplitude of oscillation, m radial axis a half of radial grid, m
ratio of e!ective ba%e area to tube area strain rate, /s energy dissipation per unit mass of #uid, W/kg kinematic viscosity of #uid, m/s density of #uid, kg/m variance of strain rate, /s angular frequency of oscillation, rad/s
eddy acoustic model relates the frictional resistance to the acoustic resistance of a single ori"ce in a thin plate and assumes that the eddy kinematic viscosity is a function of the oscillation frequency and a &mixing length' corresponding to the average distance travel of turbulent eddies. The power input for this model appears to be justi"ed for conditions of low amplitude and high oscillation frequency (1}5 mm, 3}14 Hz), giving the range of the oscillatory Reynolds number from 942 to 21 991, and the range of the Strouhal number from 3.98 to 0.8. The range of the oscillation amplitudes and frequencies tested in our work was 2}8 mm and 0.4}1.6 Hz, * the corresponding ranges of the oscillatory Reynolds number and the Strouhal number are from 251 to 4021 and from 1.99 to 0.5, respectively * which falls to the quasi-steady power model, and thus the power input per unit volume of an OBC can be estimated from P 2oN 1!a " xu, (3) < 3pC a " where N is the number of ba%es per unit length (/m), a the ba%e free area ratio ("(D /D)) where D is the
ori"ce diameter (m), o the density of the liquid and C the " ori"ce discharge coe$cient (taken as 0.7). For a given ba%e geometry, the power input in an OBC is proportional to the cube of the oscillatory velocity, x f. OBCs are being increasingly exploited in Chemical Engineering unit operations in the pursuit of process enhancements, such as in heat transfer (Mackley, Tweedle & Wyatt, 1990; Mackley & Stonestreet, 1995), mass transfer (Hewgill, Mackley, Pandit & Pannu, 1993; Ni, Gao, Cumming & Pritchard, 1995a,b), particle mixing and separation (Mackley, Smith & Wise, 1993), liquid-phase reaction (Ni & Mackley, 1993), polymerisation (Ni, Zhang & Mustafa, 1998, 1999), #occulation (Gao, Ni, Cumming, Greated & Norman, 1998) and crystallisation. The ability to fully understand and model such a #ow at a fundamental level is crucial to the success of these operations. Quantitative measurement of #uid kinematics, e.g. velocity and strain rate, and its use to understand transport phenomena is fundamental to this demand. Numerical simulation and modelling of oscillatory #ow in a ba%ed tube have been carried out previously, see for example, Howes (1988), Howes, Mackley & Roberts (1991), Mackay, Mackley & Wang (1991), Roberts (1992). This was carried out using a "nite-di!erence axi-symmetrical, time-dependent stream function plus vorticity solver. The #ow is assumed to be spatially periodic, with the #ow in each cell identical, and the vortices formed are symmetric to the centre-line of the column (Howes, 1988; Roberts, 1992). By exploiting the axi-symmetric nature of the OBC these models have successfully predicted the onset of chaotic motions, predicted concentration gradients by incorporating transport phenomena such as heat and mass transfer, and provided particle motion simulations up to a critical oscillatory Reynolds number. Depending upon the column geometry and the viscosity of #uid, the critical oscillatory Reynolds number may vary. For this work, the observed critical oscillatory Reynolds number is around about 1580. When the oscillatory Reynolds number increases beyond such a critical value, the generation of vortices is no longer axi-symmetric, and currently available simulation techniques can no longer be applied. In this study we report, for the "rst time, our direct measurement of velocity and strain rate in an OBC using the particle image velocimetry (PIV) technique. The measurements presented in this paper cover both axi-symmetric (Re )1600) and non-axisymmetric (1600(Re (4021) regimes in an OBC, and the results serve as a prelude to our #occulation study of bacteria, Alcaligenes eutrophus, in the device. 2. Experimental facilities and precedure PIV is a well-established tool for the measurement of #uid #ow and was chosen because it allows quantitative measurements over a large region of a #ow "eld. An
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indication of its widespread usage can be found in Adrian (1996). The technique is also well documented by Adrian (1991) and Ra!el, Willert and Kompenhans (1998), and therefore only a brief description of PIV applicable to our case is given below, along with a description of the OBC and the experimental procedure. 2.1. The oscillatory ba{ed column A schematic diagram of the OBC used in the studies, together with the PIV set-up as well as the measurement area, is shown in Fig. 1. The device consists of a Perspex column, 50 mm in diameter and 500 mm in length, with a liquid capacity of 1 l. Six ori"ce ba%es were made of PTFE sheet of 3 mm thick and connected by two stainless-steel threaded rods. The ba%es were spaced 75$0.5 mm apart. The base of the column was attached to a stainless-steel bellows assembly, 50 mm in diameter. This was driven by a crank mounted to a #y-wheel on the shaft of an electrical motor. The oscillation amplitudes were obtained by adjusting the eccentric distance between the crank and the #y-wheel. The speed of the motor could be varied to provide oscillation frequencies between 0.2 and 10 Hz with increments of 0.1 Hz. Eight foil strips were equally spaced around the circumference of the #y-wheel and detected by a photodiode. This provided eight trigger pulses, indicating the exact phase of oscillation within each cycle. This is an important experimental element, which allows us to examine the detailed variations of the velocity and strain rate within a cycle of #uid oscillation. A rectangular Perspex box was "tted around the OBC at the measurement area and "lled with tap water, removing geometrical distortion, which would have otherwise been present due to the circular geometry of the column. 2.2. Illumination and optics The light source was a 4 W, continuous-wave, ArgonIon laser (Spectra Physics 2017). The output beam was collimated and directed onto an octagonal rotating mirror. The rotation of the mirror caused the beam to be swept repeatedly through an arc and onto a parabolic mirror. This, in turn, directed the laser beam horizontally through the OBC, producing a pseudo light sheet, 500 mm long and 2 mm thick in the #ow "eld, as shown in Fig. 1. The area under investigation is a half-ba%ed cell covering 50 mm across the column and 36 mm along (Fig. 1). A second photodiode was located outside the scanning beam box to detect each sweep of the laser beam. This was synchronised with the output from the "rst photodiode that monitors the phase of oscillation. The combination of the two signals enables us to control the velocity measurement so that the starting phase of oscillation over a cycle always corresponds to the start of the beam scan. Tracer particles used in the measurement
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were neutrally buoyant hollow glass spheres with a mean particle diameter of 12 lm. 2.3. Recording A purpose-built camera system, designed and manufactured by the University of Edinburgh, was located perpendicular to the light illumination sheet to record the side-scattered light from the seeding particles. The camera unit incorporated two Pulnix TM-9701 CCD cameras, a beam splitting cube and translation mechanisms for the CCD arrays. The light from the seeding particles passed through a 55 mm Micro-Nikkor lens of f1:2.8 into the beam splitting cube. This directed the incoming light onto the CCD arrays, with each receiving 50% of the incoming light. A detailed technical description of the camera system can be found elsewhere (Dewhirst, 1998). The camera exposure was controlled electronically from a separate box synchronising the pulses from both photodiodes described previously. Once both signals had been received, the exposure pulses were forwarded to the cameras so that each camera was exposed during a single sweep of the laser beam along the parabolic mirror, the second camera being exposed after the "rst at a controlled separation time. The images from the tracer particles were thus recorded at di!erent positions in the two arrays. Cross-correlating the two images yielded the velocity vectors in the plane of the pseudo light sheet. The rotational speed of the octagonal mirror can be varied from 50 to 250 rps, which corresponds to a scanning rate of 400}2000 Hz. The sweep time of the laser beam can be set with an accuracy of $0.1 ms; the minimum scan time (0.5 ms) governs the minimum time separation between the two consecutive camera exposures, which also sets the upper-limit of measuring capability for the current PIV system. The video signals from each camera's analogue output were fed into separate frame grabbing boards (Data Translation 3155) at a transmission rate of around about 3 Mb data/s and saved to disks via two PC computers. 2.4. Experimental procedure A litre of water was poured into the OBC and tracer particles added to give a seeding to liquid ratio of 10\ : 1, recommended by Elgobashi (1994) as the optimal seeding density. For each run the seeded #uid was allowed to oscillate for 5 min before image acquisition took place. Water temperature was also recorded before and after each run in order to compensate from any temperature e!ects that may occur. The experiments were carried out for three oscillation amplitudes of 2, 4 and 8 mm and four oscillation frequencies of 0.4, 0.8, 1.2 and 1.6 Hz, resulting in twelve amplitude}frequency combinations. The oscillatory Reynolds number varied between 251 and 4021, covering both the axi-symmetric
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Fig. 1. A schematic diagram of the OBC, PIV image acquisition system and the measurement area.
(Re )1600) and non-axisymmetric (1600(Re (4021) regimes for the given geometry of the column. For each individual experiment, 15 images were taken at each of the eight equal phases of oscillation over a cycle, totalling 120 images per cycle. To test the repeatability of experiments, each combination of oscillation condition was performed twice, each time acquiring 120 images.
the displacement has been computed, it was then scaled by the known time separation to yield the required velocities. The shear strain rate, c, was calculated using the formula:
2.5. Data analysis
where the co-ordinates of x and y correspond to the axial (along the column) and the radial (across) axis as indicated in Fig. 1. In this respect, u denotes the axial velocity component and v the radial velocity component. In the following presentation, three catalogues of data are reported. Firstly, instantaneous velocities and strain rates were obtained by interrogating single image-pair at
Velocity vector maps for each image-pair were produced by interrogating small sub-regions (known as interrogation areas) of the images and performing a statistical spatial correlation routine which determined the average displacement between particles within that area. Once
u !u v !v G H\ # G> H G\ H , c(i, j)" G H> y !y x !x G H> G H\ G> H G\ H
(4)
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
each phase of oscillation within a cycle. Such instant properties are the snap shots of what is going on during the investigation. Secondly, averaged properties were computed, associated with the phase-averaged velocities and strain rates, i.e. averaging 15 images for each of the eight phases of oscillation over a cycle. This provides an aggregated-mean picture linking the velocities and strain rates with the phase position of oscillation. Finally, global properties, including both cycle and volume-averaged measurements, are also calculated. Those quantities emphasis the overall intensity within the half-ba%ed cell under investigation. The instantaneous and phase/cycleaveraged analyses are time-domain processes while the global means are calculated in terms both time and spatial co-ordinates. The scope of data analyses comprises several thousand velocity and strain rate maps involving interrogations of some several million individual data points. In this paper, we present only a small selection of the results. 3. Results and discussion 3.1. Instantaneous velocity and strain rate proxles Figs. 2}9 show one of the typical strain rate distributions (in colour) superimposed with the velocity vectors for eight phases over a single oscillation cycle, respectively. The oscillation frequency was 1.6 Hz, and the amplitude was 4 mm centre-to-peak, which corresponds to Re "2011 and St"0.995. The two black boxes just above the radial axis, >, indicate the position of the ori"ce ba%e present. Note that due to the nature of PIV data analysis we cannot extend our calculations close to ba%es and walls in the measurement area, as a result, there existed small gaps between the strain rate/velocity graphs and the ba%e as well as the axis. This, however, does not a!ect data interpretation. The reference velocity vector of 150 mm/s shown in the centre of the ba%e gives an indication of the magnitude of the oscillatory velocity experienced in the column. In order to show the details of both the strain rate and velocity variations we follow through the development over an entire cycle. Fig. 2 shows the #ow pattern at the beginning of an upstroke. There is no obvious mean #ow in the axial or radial directions and the levels of strain rate are relatively low (predominately green colour). Progressing through the cycle on the upstroke (Figs. 3 and 4) a mean axial velocity through the centre of the tube becomes evident. This results in the formation of a single toroidal vortex (visually two vortices) in the section just above the ba%e (Fig. 4) with two small areas of high shear (orange yellowish colour on one side, blue and navy blue colour on another) noticeable just above the inner diameter of the ba%e. Further through the cycle (Figs. 5 and 6), it is apparent that the highest strain rates are found almost in phase with the generation of vortices
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and that the levels of strain rate increase from the beginnings of vortex formation to the vortex being fully developed (Fig. 5). It is also evident that the two vortex sections are of a non axi-symmetrical nature (Fig. 6). On #ow reversal (Fig. 7) the vortex is fully shed from the ba%e and progressed upwards, and the regions of relatively high shear become spatially larger, although the levels of the maximum shear decrease in magnitude. At 6/8th way through the cycle in the downstroke (Fig. 8) a mean downward #ow can be observed along the sides of the column constricting the diameter of the vortices while the cross-sectional diameters still increase. The levels of strain rate associated with the vortices at this stage are decreasing in both magnitude and spatial extent. As we progress further through the downward stroke the vortices continue to convect in a non-axisymmetrical manner followed by further reduced strain rate (Fig. 9). This sequence of the eight graphs provides only a snap shot of the variations of velocity and strain rate as a function of time during the investigation, and represents merely 8 out of 2880 similar pro"les at di!erent operational conditions. The graphs shown here are highly reproducible. The details of all the graphs, including those not presented in the paper, outline two key issues. Firstly, the #ow patterns (velocity vector maps) at the axi-symmetrical regime (Re 1600) qualitatively agree with the previous experimental (Brunold et al., 1989; Dickens et al., 1989; Mackley & Ni, 1991) and numerical (Howes, 1988; Howes et al., 1991; Mackay et al., 1991; Roberts, 1992) observations, which validates our PIV results. Secondly the PIV technique has also, for the "rst time, provided detailed information in relation to velocity and strain rate beyond the axi-symmetrical regime where no numerical methods have yet managed to do so. The PIV technique thus o!ers us a step change in ability to perform advanced #uid #ow modelling, and enables us to study #ow regimes previously not amenable to quantitative analysis and to obtain physically based explanations at the microscopic level for macroscopically observed phenomena. 3.2. Phase and cycle-averaged properties As previously stated, the phase-averaged properties were obtained by averaging 15 images for each individual phase of an oscillation cycle. Figs. 10}17 show such velocity vector maps for an oscillation frequency of 1.6 Hz, and amplitude of 8 mm centre-to-peak, which corresponds to Re "4021 and St"0.497. The reference velocity vector shown in the centre of the ba%e is 300 mm/s in magnitude. It can be seen that at the start of the cycle (Fig. 10) there is little obvious mean #ow in either axial or radial direction, and the magnitude of the velocity vectors is predominately small. Moving on in the cycle, a single toroidal vertex (visually a pair of vortices)
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Fig. 2. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). Start of a cycle (i.e. bottom of piston stroke).
Fig. 3. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 1/8th way through cycle on upstroke.
is about to be generated just above the ba%e in Fig. 11, is formed in Fig. 12 and progressed upwards in Figs. 13 and 14. At the same time a mean upward #ow becomes stronger. At the beginning of the downward stroke (Fig. 15) the vortices convect radially, resulting in an
increase in the magnitude of the radial velocity component and a mean downward #ow along the walls of the column. Completing the cycle of oscillation (Figs. 16 and 17), the downward #ow becomes a dominate feature coupled with the decrease in magnitude.
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Fig. 4. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 1/4th way through cycle on upstroke.
Fig. 5. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 3/8th way through cycle on upstroke.
Following through the sequence of oscillation, we note that the characteristics of the phase-averaged velocity vectors are generally similar to those of the instantaneous velocity maps, but the axi-symmetrical nature is much more evident. This is perhaps due to the nature of the oscillatory motion, and some of the extreme #uctuations have been cancelled out during the averaging process. By averaging the phase-averaged velocity and strain rate over a cycle of oscillation, we obtained the cycleaveraged properties. Figs. 18 and 19 display such cycleaveraged velocity and strain rate, respectively. Initially,
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Fig. 6. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). Halfway through cycle (i.e. top of upstroke).
Fig. 7. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 5/8th way through cycle on downstroke.
we thought that performing a cycle average over the phase-averaged data would produce a zero mean velocity vector map since the positive velocity vectors are equal in magnitude and opposite in direction with the negative ones, but the results showed a di!erent picture, which is consistent for all the experimental data obtained (2880 in total). We understand that the time-averaged #ow "eld is associated with the non-linear nature of the Navier}Stokes equation. This non-linearity due to the inertial e!ects induces the di!erence between the streamlines on the upstroke and downstroke, leading to the
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Fig. 8. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 6/8th way through cycle on downstroke.
Fig. 9. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 7/8th way through cycle on downstroke.
time-averaged inertial structure shown in Fig. 18, i.e. the steady streaming phenomena. Such phenomena can also take place in the PIV analysis. By closely examining the steady streaming structure in the OBC, we have learnt that the eddy motion generated in situ within the OBC not only enhances the radial motion, producing similar magnitudes of velocity components in both axial and radial directions, but also results a net vortex convection after each cycle of oscillation. That means that on the completion of every cycle of oscillation particles/#uids have convected both across and along the column. It is such a net convective motion
Fig. 10. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). Start of a cycle (i.e. bottom of piston stroke).
Fig. 11. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 1/8th way through cycle on upstroke.
that is responsible for mixing particles/#uids evenly throughout the column. Particles/#uids at the top of the column for example are constantly reaching all parts of the column, and vice versa. The results in Fig. 18 are signi"cant as they show for the "rst time the transport phenomena or convective motion of eddy experienced at each cycle of oscillation within such a column. Fig. 19 shows the cycle-averaged strain rate map at an oscillation frequency of 1.6 Hz and amplitude of 4 mm (Re "2011 and St"0.995), onto which a cycle-averaged vector map similar to that shown in Fig. 18 was superim-
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Fig. 12. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 1/4th way through cycle on upstroke.
Fig. 13. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 3/8th way through cycle on upstroke.
posed. Note that the scale of the strain rate in Fig. 19 is from 0 to 15/s, instead of from !7.5 to #7.5/s, for the convenience of calculations by taking the modulus of the strain rate values. It can be seen that the high level of the strain rates coincides with the net eddy convection after the cycle of oscillation, which justi"es our methods of calculations. This also suggests that an aggregated relatively high level of strain rate shall always accompany the net vortex convection after each cycle of oscillation. The cycle-averaged velocity and strain rate maps in Figs. 18 and 19 give a visual impression of more #ow in
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Fig. 14. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). Halfway through cycle (i.e. top of upstroke).
Fig. 15. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 5/8th way through cycle on downstroke.
the centre of the column than the wall. This is in fact not true, as the cycle-averaged properties are not radially weighted, i.e. the cycle-averaged values are the mean velocity of the instantaneous velocity measurements over a cycle of oscillation. When the cycle-averaged velocity and strain rate maps have gone through a volume average procedure, such an arti"cial impression is eliminated. In addition, the continuity equation ((*u/*x)#(*v/*y)"0) has been satis"ed in our calculations for almost all the transient velocity points at any given cross section of the OBC. There was, however, a very small proportion of data that were of a non-zero nature, and this is caused,
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Fig. 16. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 6/8th way through cycle on downstroke.
Fig. 18. Cycle averaged the phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz).
Fig. 17. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 7/8th way through cycle on downstroke.
Fig. 19. Cycle averaged the phase-averaged strain rate map (Re "2011, x "$4 mm, f"1.6 Hz).
we believe, by the existence of a third component in the #ow in our measurement.
respectively, as
3.3. Global characteristics Both the instantaneous properties and phase/cycleaveraged properties described above are the time-domain velocity and strain rate calculations. Global characteristics of the measured properties include both the time and spatial elements, and two quantities used to meet this demand are the volume-averaged strain rate, c , and the variance of the strain rate, p, de"ned,
1 c " "c(i, j)y "*x *y, (5) (x !x )y H G H 1 p" ["c(i, j)"!c ]y *x *y, (6) H (x !x )y G H where y is a half of the radial grid in the measurement area and (x !x ) the length of the axial grid. In (5), the mean of the absolute strain rate is calculated as this overcomes the tendency of the negative and positive strains being cancelled out. The variance of the strain
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rate in (6) gives an indication of the extent of the strain rate #uctuation within the measurement region. Fig. 20 shows the volume-averaged absolute strain rate plotted against the phase of oscillation for four oscillation frequencies of 0.4, 0.8, 1.2 and 1.6 Hz at a "xed oscillatory amplitude of 8 mm (St"0.497), which was also the maximum amplitude tested. The corresponding oscillatory Reynolds numbers are 1005, 2011, 3016 and 4021, respectively. Note that there are two data series plotted for each frequency corresponding to two individual experimental runs under the same operational conditions. This provides the degree of the repeatability in the data analysis. The paired curves at each of the four frequencies generally show the same trends; relatively low values of shear at the beginning of the cycle which progressively increase to a well-de"ned maximum before decreasing to shear values similar to those at the start of the cycle. The graph clearly indicates that the strain rate peaks at the stage of the cycle, where the vortices have fully formed above the ba%e as shown in Figs. 5 and 13. The volume-averaged strain rate graphs for oscillation amplitudes of 4 mm (St"0.995) (Fig. 21) and 2 mm (St"1.989) (Fig. 22) display similar characteristics but with much reduced strain rate. The consistency of the maximum peak of the strain rate for all the experiments carried out also justi"es the data measurement and data analysis techniques employed in the study. It should, however, be noted that the current design of the experimental apparatus imposes a constraint on the ability to obtain valid velocity results close to the ba%es. This most likely a!ected the strain rate results for the low oscillatory amplitude (2 mm) more than the higher ones, which can clearly be seen in Fig. 22 where the maximum peak of the strain rate becomes less well de"ned. The corresponding plots of the variance of the strain rate vs. the phase of oscillation for oscillation amplitudes of 8, 4 and 2 mm (St"0.497, 0.995 and 1.989) are given in Figs. 23}25, respectively. Clearly, the variance of the strain rate generally follows the same pattern as the mean strain rate, which is expected. The maximum #uctuation of the variance of the strain rate also peaks at the stage of oscillation with a deviation at the lowest oscillation amplitude of 2 mm for the same reasons. The variance of the mean strain rate in fact gives a measure of the peak strain rate experienced in such a column. Such peak strain rates, not the average rates, are very important in biochemical and biomedical applications as they govern the extent of cell damage caused due to the shear e!ect. Table 1 summaries the peak values of both the volumeaveraged strain rates and the variance for a range of oscillatory Reynolds numbers tested in this work and the values quoted in Table 1 are the average of the two sets of data plotted in Figs. 20 to 25. We should state that such peak values are more accurate for the higher oscillatory Reynolds numbers (*1005) than the lower ones for the same reasons mentioned previously. It is clear that in
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Fig. 20. Volume-averaged strian rate vs. phase of oscillation. St"0.497 (8 mm).
Fig. 21. Volume-averaged strian rate vs. phase of oscillation. St"0.995 (4 mm).
spite of the given appreciable spread of the strain rate in the OBC, the peak strain rates are very small. By averaging the strain rate data shown in Figs. 20}22 over the cycle, we obtain the overall mean strain rate. Fig. 26 plots the overall mean strain rate against the oscillatory Reynolds numbers. It can be seen that the overall mean strain rate varies almost linearly with the oscillatory Reynolds number. The signi"cance of this graph is that it provides, for the "rst time, quantitative values for the average strain rate in an OBC for a range of oscillation conditions.
4. Correlation between strain rate and power dissipatoin The estimation of strain rate in other systems is generally empirical and associated with either mixing intensity
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Fig. 22. Volume-averaged strian rate vs. phase of oscillation. St"1.989 (2 mm).
Fig. 24. Variance of strian rate vs. phase of oscillation. St"0.995 (4 mm).
Fig. 23. Variance of strian rate vs. phase of oscillation. St"0.497 (8 mm).
Fig. 25. Variance of strian rate vs. phase of oscillation. St"1.989 (2 mm).
or power dissipation. In this work, we are particularly interested in establishing such a correlation linking the mean strain rate with the oscillatory Reynolds number, and with the power dissipation in the OBC. Based on the large number of experiments carried out in the OBC, the following correlation, as shown in Fig. 26, was obtained as
Combining Eqs. (7) and (8), the mean strain rate in the OBC can be expressed in terms of e as
c"6;10\ (Re )
(/s) 251)Re )4021,
(7)
which relates the mean strain rate with the oscillatory Reynolds number. The correlation coe$cient is of 98%. In order to relate the mean strain rate with the average energy dissipation per unit mass, e"P/
(8)
c"42.6 e (/s).
(9)
This quantitative correlation is the "rst of its kind in an OBC linking the mean strain rate with the power dissipation. It clearly shows that the mean strain rate is controlled by the power input to the OBC, and using the PIV technique we can quantify the strain rate exerted in the column simply by calculating the power input. 5. Comparison with stirred tank vessels A direct comparison of shear rate between an OBC and a stirred tank (ST) vessel is di$cult due to the nature of the system, the comparison carried out here is merely on an approximate basis. In order to compare the strain rate, we need to choose a common base for the two systems. Metzner and Otto in 1957 proposed that the
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Table 1 The peak values of the volume-averaged strain rates and the variance in the OBC Re x (mm) f (Hz) St (c ) (/s) 4 (p) (/s)
4021 8 1.6 0.497 21.4
3016 8 1.2 0.497 17.5
2011 8 0.8 0.497 11.1
1508 4 1.2 0.995 6.4
1005 8 0.4 0.497 5.1
754 2 1.2 1.989 2.3
503 2 0.8 1.989 1.3
251 2 0.4 1.989 0.7
38.7
30.1
19.5
11.0
8.3
5.0
4.2
1.4
Table 2 A comparison of the mean strain rate in an OBC and STs N (rps)"f (Hz) 0.4 (24 rpm) 0.8 (48 rpm) 1.2 (72 rpm) 1.6 (96 rpm) c (/s) 12 c (/s) - ! c (Oldshue, 12 1983)
Fig. 26. Volume and cycle-averaged strian rate vs. oscillatory Reynolds number.
average shear rate in a ST (c ) is directly proportional to 12 the agitation speed as c "kN, (11) 12 where N is the rotational speed (rpm) and k a constant which depends on impeller design and the type of #uid. Due to its empirical nature there have been a great number of correlation for the k value, e.g. (Calderbank & Moo-Young, 1961) k"B
L \L 4n , 3n#1
(12)
where n is the index of the power-law #uids, in this case, pseudo-plastic #uid and B a system constant that they found to be 11. For a 6-bladed disc turbine, 2 bladed paddles and 3 bladed propeller, Harnby, Edwards & Nienow (1992) found that k"11.5, 10 and 10, respectively. The Metzner}Otto relationship is valid for low Reynolds number #ows in STs, and not for fully turbulent #ows. As our work also belongs to the low Reynolds number #ow regimes, this is in fact the identi"ed common ground for comparison. We decided to take an average value of k"10.5 and apply it to the Metzner} Otto relationship for the same range of revolution per second or oscillation frequency applied. Table 2 shows such a comparison for both systems. In addition, we have
4.2 3.5 No data
8.4 7.1 16.4
12.6 10.1 26.5
16.8 13.3 31.1
also quoted the mean shear data of a 6-in six-blade turbine impeller rotating in a 18-in ST containing water (Oldshue, 1983) for reference. Note that the oscillation amplitude of 8 mm was selected for the comparison, as it was the largest amplitude in our investigation. It is evident that for the same revolution per second the mean strain rate in the OBC is lower than that in a ST vessel estimated using the Metzner}Otto relationship, bear in mind that at such rotational speeds there will be little mixing in a given ST vessel. It is also clear that the mean strain rate in our 50-mm diameter OBC is much smaller than that in a 457-mm diameter ST vessel. This is very encouraging news as the larger the ST vessel, the smaller the strain rate, and the strain rate in our system is more favourable compared with the scale-up ST. This result is particularly important and relevant as OBCs could be applied to biochemical and biomedical "elds involving cell cultures, and a good global mixing complemented with very low shear rate is essential to such processes.
6. Conclusions The PIV has provided quantitative time-resolved information of velocity and strain rate in an OBC. The instantaneous, phase/cycle-averaged properties together with the global characteristics provide a deep insight into the #uid mechanical conditions, the mixing and convective transport phenomena in both axi-symmetric and non-axisymmetric regimes. The quantitative data has opened up a new world of analysis of #uid #ow
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behaviour and transport phenomena in OBCs, which other methods are unable to provide. For the range of the oscillatory Reynolds numbers (251}4021) investigated the volume-averaged mean strain rate varies from 0.5 to 13.5/s, respectively, which is lower than ST vessels. This is particularly important as it suggests that OBCs could be applied to biochemical and biomedical "elds where cultures are involved, and a good global mixing complemented with very low shear rate is essential to such processes. As a prelude to our #occulation study, the results shown in this paper are very encouraging as #oc formation and growth in #occulation processes are sensitive to the magnitude of the strain rate. The potential to simultaneously attain good global mixing and low shears are highly desirable. We have also, for the "rst time, presented the quantitative correlation between the mean strain rate and the power dissipation in an OBC. Using the PIV technique we are able to quantify the strain rate, no longer relying on empirical estimations.
Acknowledgements Authors wish to thank BBSRC for the "nancial support of this project.
References Adrian, R. J. (1991). Particle-imaging techniques for experimental #uid mechanics. Annual Review of Fluid Mechanics, 23, 261}304. Adrian, R. J. (1996). Bibliography of particle velocimetry using imaging methods: 1917}1995. Technical Report 817, USA: University of Illinois. Baird, M. H. I., & Stonestreet, P. (1995). Energy dissipation in oscillatory #ow within a ba%ed tube. Transactions of the Institute of Chemical Engineers, 73, 503}511. Brunold, C. R., Hunns, J. C. B., Mackley, M. R., & Thompson, J. W. (1989). Experimental observations on #ow patterns and energy losses for oscillatory #ows in ducts with sharp edges. Chemical Engineering Science, 44, 1227}1244. Calderbank, P. H., & Moo-Young, M. (1961). Transactions of the Institute of Chemical Engineers, 39, 337}342. Dewhirst, T.P. (1998). Particle-imaging techniques for experimental yuid mechanics. Ph.D. Thesis, Scotland: The University of Edinburgh. Dickens, A. W., Mackley, M. R., & Williams, H. R. (1989). Experimental residence time distribution measurements for unsteady #ow in baf#ed tubes. Chemical Engineering Science, 44, 1471}1479. Elgobashi, S. (1994). On predicting particle-laden turbulent #ows. Applied Science Research, 52, 309}329.
Gao, S., Ni, X., Cumming, R. H., Greated, C. A., & Norman, P. (1998). Experimental investigation of bentonite #occulation in a batch oscillatory ba%ed column. Separation Science and Technology, 33, 2143}2157. Harnby, N., Edwards, M. F., & Nienow, A. W. (1992). Mixing in the process industries (2nd Ed). London: Butterworth-Heinemann Ltd. Hewgill, M. R., Mackley, M. R., Pandit, A. B., & Pannu, S. S. (1993). Enhancement of gas}liquid mass transfer using oscillatory #ow in a ba%ed tube. Chemical Engineering Science, 48, 799}809. Howes, T. (1988). On the dispersion of unsteady yow in ba{ed tubes. Ph.D. thesis, Cambridge University. Howes, T., Mackley, M. R., & Roberts, E. P. L. (1991). The simulation of chaotic mixing and dispersion for periodic #ows in ba%ed channels. Chemical Engineering Science, 46, 1669}1677. Mackay, M. E., Mackley, M. R., & Wang, Y. (1991). Oscillatory #ow within tubes containing wall or central ba%es. Transactions of the Institute of Chemical Engineers, 69(A), 506}513. Mackley, M. R., & Ni, X. (1991). Mixing and dispersion in a ba%ed tube for steady laminar and pulsatile #ow. Chemical Engineering Science, 46, 3139}3151. Mackley, M. R., & Ni, X. (1993). Experimental #uid dispersion measurements in periodic ba%ed tube arrays. Chemical Engineering Science, 48, 3293}3305. Mackley, M. R., Smith, K. B., & Wise, N. P. (1993). The mixing and separation of particle suspensions using oscillatory #ow in ba%ed tubes. Transactions of the Institute of Chemical Engineers, 71(A), 649}656. Mackley, M. R., & Stonestreet, P. (1995). Heat transfer and associated energy dissipation for oscillatory #ow in ba%ed tubes. Chemical Engineering Science, 50, 2211}2224. Mackley, M. R., Tweedle, G. M., & Wyatt, I. D. (1990). Experimental heat transfer measurements for pulsatile #ow in ba%ed tubes. Chemical Engineering Science, 45, 1237}1242. Metzner, A. B., & Otto, R. E. (1957). A.I.Ch.E. Journal, 3, 13}20. Ni, X., Gao, S., Cumming, R. H., & Pritchard, D. W. (1995a). A comparative study of mass transfer in yeast for a batch pulsed ba%ed bioreactor and a stirred tank fermenter. Chemical Engineering Science, 50, 2127}2136. Ni, X., Gao, S., & Pritchard, D. W. (1995b). A study of mass transfer in yeast in a pulsed ba%ed bioreactor. Biotechnology and Bioengineering, 45, 165}175. Ni, X., & Mackley, M. R. (1993). Chemical reaction in batch pulsatile #ow and stirred tank reactors. Chemical Engineering Journal, 52, 107}114. Ni, X., Zhang, Y., & Mustafa, I. (1999). Correction of polymer particle size with droplet size in suspension polymerisation of methylmethacrylate in a batch oscillatory ba%ed reactor. Chemical Engineering Science, 54, 841}850. Ni, X., Zhang, Y., & Mustafa, I. (1998). An investigation of droplet size and size distribution in methylmethacrylate suspensions in a batch oscillatory ba%ed reactor. Chemical Engineering Science, 53, 2903}2919. Oldshue, J. Y. (1983). Genetic engineering news (pp. 46}49). Ra!el, M., Willert, C., & Kompenhans, J. (1998). Particle image velocimetry. Berlin: Springer. Roberts, E. P. L. (1992). Unsteady yow and mixing in ba{ed channels, Ph.D. thesis. Cambridge University.
On the measurement of strain rate in an oscillatory ba%ed column using particle image velocimetry X. Ni *, J. A. Cosgrove , A. D. Arnott, C. A. Greated, R. H. Cumming Department of Mechanical and Chemical Engineering, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, UK Department of Physics and Astronomy, The University of Edinburgh, Edinburgh, Scotland, EH9 3JZ, UK Applied Sciences Section, The University of Teesside, Middlesborough, TS1 3BA, UK Received 21 January 1999; accepted 13 September 1999
Abstract We report, for the "rst time, our direct experimental measurement of velocity vectors and strain-rate distributions in an oscillatory ba%ed column (OBC) using time-resolved particle image velocimetry (PIV). The technique allowed a time series of spatial velocity maps to be obtained for several phases per oscillation cycle and the results, comprising several thousand such maps, illustrate in detail the variations of velocity and strain rate in a ba%ed region over a range of oscillation amplitudes and oscillation frequencies and provide a deep insight into the mixing, vortex convection and transport mechanism in such a device. The results show that an OBC not only provides enhanced mixing, but also o!ers low strain rates, which are lower than those in stirred tank vessels. We have also reported for the "rst time the quantitative correlation between the strain rate and the power dissipation in an OBC. Using the PIV technique we are able to quantify the strain rates experienced in an OBC. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Oscillatory ba%ed column; Particle image velocimetry; Velocity vector; Strain rate; Vortices; Oscillation frequency; Oscillation amplitude
1. Introduction Many #ows present signi"cant challenges to experimental measurement, analysis and modelling, especially those where the motion is very complex in both space and time domains. A typical example is oscillatory (unsteady) #ow in a ba%ed tube. In such a device, #uid mixing is achieved by eddies that are generated when #uid passes through a set of equally spaced stationary ori"ce ba%es. Those periodically formed vortices can be controlled by a combination of geometrical and operational parameters, such as, ba%e diameter, ba%e spacing, oscillation frequency and oscillation amplitude. Under certain operational conditions, an oscillatory ba%ed column (OBC) can be operated as either a plug-#ow reactor or an enhanced mixing device (Brunold, Hunns, Mackley & Thompson, 1989; Dickens, Mackley & Williams, 1989; Mackley & Ni, 1991,1993). For a given ba%e geometry, the #uid mechanical condition in an OBC is controlled by the oscillatory Reynolds number, Re , and the * Corresponding author. Tel.: #0131-451-3781x4723; fax: #0131451-3077. E-mail address: [email protected] (X. Ni)
Strouhal number, St, de"ned as Dx u Re " , l
(1)
D St" , (2) 4px where D is the column diameter (m), x the oscillation amplitude (m), u the angular frequency of oscillation ("2pf ), f is the oscillation frequency (Hz), and l the kinematic viscosity of #uid (m/s). There are essentially two models for estimating the power input in an OBC: the quasi-steady #ow model and the eddy acoustic model (Baird & Stonestreet, 1995). The former assumes that oscillatory #ows are &quasi-steady', i.e. the instantaneous pressure drop in a time-periodic #ow is assumed to be identical to the pressure drop that would be obtained at a steady velocity of the same magnitude of the instantaneous velocity. The power input for this model is valid for higher oscillation amplitudes and lower frequencies, e.g. 5}30 mm, 0.5}2 Hz. For the same scale of the column and the same viscosity of #uid as used in this work, this corresponds to the range of the oscillatory Reynolds number from 785 to 18 850 and the range of the Strouhal number from 0.8 to 0.13. The
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 5 7 7 - 1
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Nomenclature B C " D D k n N N P/< Re St u v x x y y Greek letters a c e l o p u
constant ori"ce discharge coe$cient ("0.7) column diameter, m ori"ce diameter, m constant index of power-law #uid number of ba%es per unit length, /m speed of stirrer, r.p.s. power density, W/m oscillatory Reynolds number de"ned by Eq. (1) Strouhal number de"ned by Eq. (2) radial velocity component, m/s axial velocity component, m/s axial axis amplitude of oscillation, m radial axis a half of radial grid, m
ratio of e!ective ba%e area to tube area strain rate, /s energy dissipation per unit mass of #uid, W/kg kinematic viscosity of #uid, m/s density of #uid, kg/m variance of strain rate, /s angular frequency of oscillation, rad/s
eddy acoustic model relates the frictional resistance to the acoustic resistance of a single ori"ce in a thin plate and assumes that the eddy kinematic viscosity is a function of the oscillation frequency and a &mixing length' corresponding to the average distance travel of turbulent eddies. The power input for this model appears to be justi"ed for conditions of low amplitude and high oscillation frequency (1}5 mm, 3}14 Hz), giving the range of the oscillatory Reynolds number from 942 to 21 991, and the range of the Strouhal number from 3.98 to 0.8. The range of the oscillation amplitudes and frequencies tested in our work was 2}8 mm and 0.4}1.6 Hz, * the corresponding ranges of the oscillatory Reynolds number and the Strouhal number are from 251 to 4021 and from 1.99 to 0.5, respectively * which falls to the quasi-steady power model, and thus the power input per unit volume of an OBC can be estimated from P 2oN 1!a " xu, (3) < 3pC a " where N is the number of ba%es per unit length (/m), a the ba%e free area ratio ("(D /D)) where D is the
ori"ce diameter (m), o the density of the liquid and C the " ori"ce discharge coe$cient (taken as 0.7). For a given ba%e geometry, the power input in an OBC is proportional to the cube of the oscillatory velocity, x f. OBCs are being increasingly exploited in Chemical Engineering unit operations in the pursuit of process enhancements, such as in heat transfer (Mackley, Tweedle & Wyatt, 1990; Mackley & Stonestreet, 1995), mass transfer (Hewgill, Mackley, Pandit & Pannu, 1993; Ni, Gao, Cumming & Pritchard, 1995a,b), particle mixing and separation (Mackley, Smith & Wise, 1993), liquid-phase reaction (Ni & Mackley, 1993), polymerisation (Ni, Zhang & Mustafa, 1998, 1999), #occulation (Gao, Ni, Cumming, Greated & Norman, 1998) and crystallisation. The ability to fully understand and model such a #ow at a fundamental level is crucial to the success of these operations. Quantitative measurement of #uid kinematics, e.g. velocity and strain rate, and its use to understand transport phenomena is fundamental to this demand. Numerical simulation and modelling of oscillatory #ow in a ba%ed tube have been carried out previously, see for example, Howes (1988), Howes, Mackley & Roberts (1991), Mackay, Mackley & Wang (1991), Roberts (1992). This was carried out using a "nite-di!erence axi-symmetrical, time-dependent stream function plus vorticity solver. The #ow is assumed to be spatially periodic, with the #ow in each cell identical, and the vortices formed are symmetric to the centre-line of the column (Howes, 1988; Roberts, 1992). By exploiting the axi-symmetric nature of the OBC these models have successfully predicted the onset of chaotic motions, predicted concentration gradients by incorporating transport phenomena such as heat and mass transfer, and provided particle motion simulations up to a critical oscillatory Reynolds number. Depending upon the column geometry and the viscosity of #uid, the critical oscillatory Reynolds number may vary. For this work, the observed critical oscillatory Reynolds number is around about 1580. When the oscillatory Reynolds number increases beyond such a critical value, the generation of vortices is no longer axi-symmetric, and currently available simulation techniques can no longer be applied. In this study we report, for the "rst time, our direct measurement of velocity and strain rate in an OBC using the particle image velocimetry (PIV) technique. The measurements presented in this paper cover both axi-symmetric (Re )1600) and non-axisymmetric (1600(Re (4021) regimes in an OBC, and the results serve as a prelude to our #occulation study of bacteria, Alcaligenes eutrophus, in the device. 2. Experimental facilities and precedure PIV is a well-established tool for the measurement of #uid #ow and was chosen because it allows quantitative measurements over a large region of a #ow "eld. An
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
indication of its widespread usage can be found in Adrian (1996). The technique is also well documented by Adrian (1991) and Ra!el, Willert and Kompenhans (1998), and therefore only a brief description of PIV applicable to our case is given below, along with a description of the OBC and the experimental procedure. 2.1. The oscillatory ba{ed column A schematic diagram of the OBC used in the studies, together with the PIV set-up as well as the measurement area, is shown in Fig. 1. The device consists of a Perspex column, 50 mm in diameter and 500 mm in length, with a liquid capacity of 1 l. Six ori"ce ba%es were made of PTFE sheet of 3 mm thick and connected by two stainless-steel threaded rods. The ba%es were spaced 75$0.5 mm apart. The base of the column was attached to a stainless-steel bellows assembly, 50 mm in diameter. This was driven by a crank mounted to a #y-wheel on the shaft of an electrical motor. The oscillation amplitudes were obtained by adjusting the eccentric distance between the crank and the #y-wheel. The speed of the motor could be varied to provide oscillation frequencies between 0.2 and 10 Hz with increments of 0.1 Hz. Eight foil strips were equally spaced around the circumference of the #y-wheel and detected by a photodiode. This provided eight trigger pulses, indicating the exact phase of oscillation within each cycle. This is an important experimental element, which allows us to examine the detailed variations of the velocity and strain rate within a cycle of #uid oscillation. A rectangular Perspex box was "tted around the OBC at the measurement area and "lled with tap water, removing geometrical distortion, which would have otherwise been present due to the circular geometry of the column. 2.2. Illumination and optics The light source was a 4 W, continuous-wave, ArgonIon laser (Spectra Physics 2017). The output beam was collimated and directed onto an octagonal rotating mirror. The rotation of the mirror caused the beam to be swept repeatedly through an arc and onto a parabolic mirror. This, in turn, directed the laser beam horizontally through the OBC, producing a pseudo light sheet, 500 mm long and 2 mm thick in the #ow "eld, as shown in Fig. 1. The area under investigation is a half-ba%ed cell covering 50 mm across the column and 36 mm along (Fig. 1). A second photodiode was located outside the scanning beam box to detect each sweep of the laser beam. This was synchronised with the output from the "rst photodiode that monitors the phase of oscillation. The combination of the two signals enables us to control the velocity measurement so that the starting phase of oscillation over a cycle always corresponds to the start of the beam scan. Tracer particles used in the measurement
3197
were neutrally buoyant hollow glass spheres with a mean particle diameter of 12 lm. 2.3. Recording A purpose-built camera system, designed and manufactured by the University of Edinburgh, was located perpendicular to the light illumination sheet to record the side-scattered light from the seeding particles. The camera unit incorporated two Pulnix TM-9701 CCD cameras, a beam splitting cube and translation mechanisms for the CCD arrays. The light from the seeding particles passed through a 55 mm Micro-Nikkor lens of f1:2.8 into the beam splitting cube. This directed the incoming light onto the CCD arrays, with each receiving 50% of the incoming light. A detailed technical description of the camera system can be found elsewhere (Dewhirst, 1998). The camera exposure was controlled electronically from a separate box synchronising the pulses from both photodiodes described previously. Once both signals had been received, the exposure pulses were forwarded to the cameras so that each camera was exposed during a single sweep of the laser beam along the parabolic mirror, the second camera being exposed after the "rst at a controlled separation time. The images from the tracer particles were thus recorded at di!erent positions in the two arrays. Cross-correlating the two images yielded the velocity vectors in the plane of the pseudo light sheet. The rotational speed of the octagonal mirror can be varied from 50 to 250 rps, which corresponds to a scanning rate of 400}2000 Hz. The sweep time of the laser beam can be set with an accuracy of $0.1 ms; the minimum scan time (0.5 ms) governs the minimum time separation between the two consecutive camera exposures, which also sets the upper-limit of measuring capability for the current PIV system. The video signals from each camera's analogue output were fed into separate frame grabbing boards (Data Translation 3155) at a transmission rate of around about 3 Mb data/s and saved to disks via two PC computers. 2.4. Experimental procedure A litre of water was poured into the OBC and tracer particles added to give a seeding to liquid ratio of 10\ : 1, recommended by Elgobashi (1994) as the optimal seeding density. For each run the seeded #uid was allowed to oscillate for 5 min before image acquisition took place. Water temperature was also recorded before and after each run in order to compensate from any temperature e!ects that may occur. The experiments were carried out for three oscillation amplitudes of 2, 4 and 8 mm and four oscillation frequencies of 0.4, 0.8, 1.2 and 1.6 Hz, resulting in twelve amplitude}frequency combinations. The oscillatory Reynolds number varied between 251 and 4021, covering both the axi-symmetric
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Fig. 1. A schematic diagram of the OBC, PIV image acquisition system and the measurement area.
(Re )1600) and non-axisymmetric (1600(Re (4021) regimes for the given geometry of the column. For each individual experiment, 15 images were taken at each of the eight equal phases of oscillation over a cycle, totalling 120 images per cycle. To test the repeatability of experiments, each combination of oscillation condition was performed twice, each time acquiring 120 images.
the displacement has been computed, it was then scaled by the known time separation to yield the required velocities. The shear strain rate, c, was calculated using the formula:
2.5. Data analysis
where the co-ordinates of x and y correspond to the axial (along the column) and the radial (across) axis as indicated in Fig. 1. In this respect, u denotes the axial velocity component and v the radial velocity component. In the following presentation, three catalogues of data are reported. Firstly, instantaneous velocities and strain rates were obtained by interrogating single image-pair at
Velocity vector maps for each image-pair were produced by interrogating small sub-regions (known as interrogation areas) of the images and performing a statistical spatial correlation routine which determined the average displacement between particles within that area. Once
u !u v !v G H\ # G> H G\ H , c(i, j)" G H> y !y x !x G H> G H\ G> H G\ H
(4)
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
each phase of oscillation within a cycle. Such instant properties are the snap shots of what is going on during the investigation. Secondly, averaged properties were computed, associated with the phase-averaged velocities and strain rates, i.e. averaging 15 images for each of the eight phases of oscillation over a cycle. This provides an aggregated-mean picture linking the velocities and strain rates with the phase position of oscillation. Finally, global properties, including both cycle and volume-averaged measurements, are also calculated. Those quantities emphasis the overall intensity within the half-ba%ed cell under investigation. The instantaneous and phase/cycleaveraged analyses are time-domain processes while the global means are calculated in terms both time and spatial co-ordinates. The scope of data analyses comprises several thousand velocity and strain rate maps involving interrogations of some several million individual data points. In this paper, we present only a small selection of the results. 3. Results and discussion 3.1. Instantaneous velocity and strain rate proxles Figs. 2}9 show one of the typical strain rate distributions (in colour) superimposed with the velocity vectors for eight phases over a single oscillation cycle, respectively. The oscillation frequency was 1.6 Hz, and the amplitude was 4 mm centre-to-peak, which corresponds to Re "2011 and St"0.995. The two black boxes just above the radial axis, >, indicate the position of the ori"ce ba%e present. Note that due to the nature of PIV data analysis we cannot extend our calculations close to ba%es and walls in the measurement area, as a result, there existed small gaps between the strain rate/velocity graphs and the ba%e as well as the axis. This, however, does not a!ect data interpretation. The reference velocity vector of 150 mm/s shown in the centre of the ba%e gives an indication of the magnitude of the oscillatory velocity experienced in the column. In order to show the details of both the strain rate and velocity variations we follow through the development over an entire cycle. Fig. 2 shows the #ow pattern at the beginning of an upstroke. There is no obvious mean #ow in the axial or radial directions and the levels of strain rate are relatively low (predominately green colour). Progressing through the cycle on the upstroke (Figs. 3 and 4) a mean axial velocity through the centre of the tube becomes evident. This results in the formation of a single toroidal vortex (visually two vortices) in the section just above the ba%e (Fig. 4) with two small areas of high shear (orange yellowish colour on one side, blue and navy blue colour on another) noticeable just above the inner diameter of the ba%e. Further through the cycle (Figs. 5 and 6), it is apparent that the highest strain rates are found almost in phase with the generation of vortices
3199
and that the levels of strain rate increase from the beginnings of vortex formation to the vortex being fully developed (Fig. 5). It is also evident that the two vortex sections are of a non axi-symmetrical nature (Fig. 6). On #ow reversal (Fig. 7) the vortex is fully shed from the ba%e and progressed upwards, and the regions of relatively high shear become spatially larger, although the levels of the maximum shear decrease in magnitude. At 6/8th way through the cycle in the downstroke (Fig. 8) a mean downward #ow can be observed along the sides of the column constricting the diameter of the vortices while the cross-sectional diameters still increase. The levels of strain rate associated with the vortices at this stage are decreasing in both magnitude and spatial extent. As we progress further through the downward stroke the vortices continue to convect in a non-axisymmetrical manner followed by further reduced strain rate (Fig. 9). This sequence of the eight graphs provides only a snap shot of the variations of velocity and strain rate as a function of time during the investigation, and represents merely 8 out of 2880 similar pro"les at di!erent operational conditions. The graphs shown here are highly reproducible. The details of all the graphs, including those not presented in the paper, outline two key issues. Firstly, the #ow patterns (velocity vector maps) at the axi-symmetrical regime (Re 1600) qualitatively agree with the previous experimental (Brunold et al., 1989; Dickens et al., 1989; Mackley & Ni, 1991) and numerical (Howes, 1988; Howes et al., 1991; Mackay et al., 1991; Roberts, 1992) observations, which validates our PIV results. Secondly the PIV technique has also, for the "rst time, provided detailed information in relation to velocity and strain rate beyond the axi-symmetrical regime where no numerical methods have yet managed to do so. The PIV technique thus o!ers us a step change in ability to perform advanced #uid #ow modelling, and enables us to study #ow regimes previously not amenable to quantitative analysis and to obtain physically based explanations at the microscopic level for macroscopically observed phenomena. 3.2. Phase and cycle-averaged properties As previously stated, the phase-averaged properties were obtained by averaging 15 images for each individual phase of an oscillation cycle. Figs. 10}17 show such velocity vector maps for an oscillation frequency of 1.6 Hz, and amplitude of 8 mm centre-to-peak, which corresponds to Re "4021 and St"0.497. The reference velocity vector shown in the centre of the ba%e is 300 mm/s in magnitude. It can be seen that at the start of the cycle (Fig. 10) there is little obvious mean #ow in either axial or radial direction, and the magnitude of the velocity vectors is predominately small. Moving on in the cycle, a single toroidal vertex (visually a pair of vortices)
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Fig. 2. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). Start of a cycle (i.e. bottom of piston stroke).
Fig. 3. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 1/8th way through cycle on upstroke.
is about to be generated just above the ba%e in Fig. 11, is formed in Fig. 12 and progressed upwards in Figs. 13 and 14. At the same time a mean upward #ow becomes stronger. At the beginning of the downward stroke (Fig. 15) the vortices convect radially, resulting in an
increase in the magnitude of the radial velocity component and a mean downward #ow along the walls of the column. Completing the cycle of oscillation (Figs. 16 and 17), the downward #ow becomes a dominate feature coupled with the decrease in magnitude.
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
Fig. 4. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 1/4th way through cycle on upstroke.
Fig. 5. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 3/8th way through cycle on upstroke.
Following through the sequence of oscillation, we note that the characteristics of the phase-averaged velocity vectors are generally similar to those of the instantaneous velocity maps, but the axi-symmetrical nature is much more evident. This is perhaps due to the nature of the oscillatory motion, and some of the extreme #uctuations have been cancelled out during the averaging process. By averaging the phase-averaged velocity and strain rate over a cycle of oscillation, we obtained the cycleaveraged properties. Figs. 18 and 19 display such cycleaveraged velocity and strain rate, respectively. Initially,
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Fig. 6. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). Halfway through cycle (i.e. top of upstroke).
Fig. 7. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 5/8th way through cycle on downstroke.
we thought that performing a cycle average over the phase-averaged data would produce a zero mean velocity vector map since the positive velocity vectors are equal in magnitude and opposite in direction with the negative ones, but the results showed a di!erent picture, which is consistent for all the experimental data obtained (2880 in total). We understand that the time-averaged #ow "eld is associated with the non-linear nature of the Navier}Stokes equation. This non-linearity due to the inertial e!ects induces the di!erence between the streamlines on the upstroke and downstroke, leading to the
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Fig. 8. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 6/8th way through cycle on downstroke.
Fig. 9. Instantaneous strain rate map superimposed with velocity vectors (Re "2011, x "$4 mm, f"1.6 Hz). 7/8th way through cycle on downstroke.
time-averaged inertial structure shown in Fig. 18, i.e. the steady streaming phenomena. Such phenomena can also take place in the PIV analysis. By closely examining the steady streaming structure in the OBC, we have learnt that the eddy motion generated in situ within the OBC not only enhances the radial motion, producing similar magnitudes of velocity components in both axial and radial directions, but also results a net vortex convection after each cycle of oscillation. That means that on the completion of every cycle of oscillation particles/#uids have convected both across and along the column. It is such a net convective motion
Fig. 10. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). Start of a cycle (i.e. bottom of piston stroke).
Fig. 11. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 1/8th way through cycle on upstroke.
that is responsible for mixing particles/#uids evenly throughout the column. Particles/#uids at the top of the column for example are constantly reaching all parts of the column, and vice versa. The results in Fig. 18 are signi"cant as they show for the "rst time the transport phenomena or convective motion of eddy experienced at each cycle of oscillation within such a column. Fig. 19 shows the cycle-averaged strain rate map at an oscillation frequency of 1.6 Hz and amplitude of 4 mm (Re "2011 and St"0.995), onto which a cycle-averaged vector map similar to that shown in Fig. 18 was superim-
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
Fig. 12. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 1/4th way through cycle on upstroke.
Fig. 13. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 3/8th way through cycle on upstroke.
posed. Note that the scale of the strain rate in Fig. 19 is from 0 to 15/s, instead of from !7.5 to #7.5/s, for the convenience of calculations by taking the modulus of the strain rate values. It can be seen that the high level of the strain rates coincides with the net eddy convection after the cycle of oscillation, which justi"es our methods of calculations. This also suggests that an aggregated relatively high level of strain rate shall always accompany the net vortex convection after each cycle of oscillation. The cycle-averaged velocity and strain rate maps in Figs. 18 and 19 give a visual impression of more #ow in
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Fig. 14. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). Halfway through cycle (i.e. top of upstroke).
Fig. 15. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 5/8th way through cycle on downstroke.
the centre of the column than the wall. This is in fact not true, as the cycle-averaged properties are not radially weighted, i.e. the cycle-averaged values are the mean velocity of the instantaneous velocity measurements over a cycle of oscillation. When the cycle-averaged velocity and strain rate maps have gone through a volume average procedure, such an arti"cial impression is eliminated. In addition, the continuity equation ((*u/*x)#(*v/*y)"0) has been satis"ed in our calculations for almost all the transient velocity points at any given cross section of the OBC. There was, however, a very small proportion of data that were of a non-zero nature, and this is caused,
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Fig. 16. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 6/8th way through cycle on downstroke.
Fig. 18. Cycle averaged the phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz).
Fig. 17. Phase-averaged velocity vector map (Re "4021, x "$8 mm, f"1.6 Hz). 7/8th way through cycle on downstroke.
Fig. 19. Cycle averaged the phase-averaged strain rate map (Re "2011, x "$4 mm, f"1.6 Hz).
we believe, by the existence of a third component in the #ow in our measurement.
respectively, as
3.3. Global characteristics Both the instantaneous properties and phase/cycleaveraged properties described above are the time-domain velocity and strain rate calculations. Global characteristics of the measured properties include both the time and spatial elements, and two quantities used to meet this demand are the volume-averaged strain rate, c , and the variance of the strain rate, p, de"ned,
1 c " "c(i, j)y "*x *y, (5) (x !x )y H G H 1 p" ["c(i, j)"!c ]y *x *y, (6) H (x !x )y G H where y is a half of the radial grid in the measurement area and (x !x ) the length of the axial grid. In (5), the mean of the absolute strain rate is calculated as this overcomes the tendency of the negative and positive strains being cancelled out. The variance of the strain
X. Ni et al. / Chemical Engineering Science 55 (2000) 3195}3208
rate in (6) gives an indication of the extent of the strain rate #uctuation within the measurement region. Fig. 20 shows the volume-averaged absolute strain rate plotted against the phase of oscillation for four oscillation frequencies of 0.4, 0.8, 1.2 and 1.6 Hz at a "xed oscillatory amplitude of 8 mm (St"0.497), which was also the maximum amplitude tested. The corresponding oscillatory Reynolds numbers are 1005, 2011, 3016 and 4021, respectively. Note that there are two data series plotted for each frequency corresponding to two individual experimental runs under the same operational conditions. This provides the degree of the repeatability in the data analysis. The paired curves at each of the four frequencies generally show the same trends; relatively low values of shear at the beginning of the cycle which progressively increase to a well-de"ned maximum before decreasing to shear values similar to those at the start of the cycle. The graph clearly indicates that the strain rate peaks at the stage of the cycle, where the vortices have fully formed above the ba%e as shown in Figs. 5 and 13. The volume-averaged strain rate graphs for oscillation amplitudes of 4 mm (St"0.995) (Fig. 21) and 2 mm (St"1.989) (Fig. 22) display similar characteristics but with much reduced strain rate. The consistency of the maximum peak of the strain rate for all the experiments carried out also justi"es the data measurement and data analysis techniques employed in the study. It should, however, be noted that the current design of the experimental apparatus imposes a constraint on the ability to obtain valid velocity results close to the ba%es. This most likely a!ected the strain rate results for the low oscillatory amplitude (2 mm) more than the higher ones, which can clearly be seen in Fig. 22 where the maximum peak of the strain rate becomes less well de"ned. The corresponding plots of the variance of the strain rate vs. the phase of oscillation for oscillation amplitudes of 8, 4 and 2 mm (St"0.497, 0.995 and 1.989) are given in Figs. 23}25, respectively. Clearly, the variance of the strain rate generally follows the same pattern as the mean strain rate, which is expected. The maximum #uctuation of the variance of the strain rate also peaks at the stage of oscillation with a deviation at the lowest oscillation amplitude of 2 mm for the same reasons. The variance of the mean strain rate in fact gives a measure of the peak strain rate experienced in such a column. Such peak strain rates, not the average rates, are very important in biochemical and biomedical applications as they govern the extent of cell damage caused due to the shear e!ect. Table 1 summaries the peak values of both the volumeaveraged strain rates and the variance for a range of oscillatory Reynolds numbers tested in this work and the values quoted in Table 1 are the average of the two sets of data plotted in Figs. 20 to 25. We should state that such peak values are more accurate for the higher oscillatory Reynolds numbers (*1005) than the lower ones for the same reasons mentioned previously. It is clear that in
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Fig. 20. Volume-averaged strian rate vs. phase of oscillation. St"0.497 (8 mm).
Fig. 21. Volume-averaged strian rate vs. phase of oscillation. St"0.995 (4 mm).
spite of the given appreciable spread of the strain rate in the OBC, the peak strain rates are very small. By averaging the strain rate data shown in Figs. 20}22 over the cycle, we obtain the overall mean strain rate. Fig. 26 plots the overall mean strain rate against the oscillatory Reynolds numbers. It can be seen that the overall mean strain rate varies almost linearly with the oscillatory Reynolds number. The signi"cance of this graph is that it provides, for the "rst time, quantitative values for the average strain rate in an OBC for a range of oscillation conditions.
4. Correlation between strain rate and power dissipatoin The estimation of strain rate in other systems is generally empirical and associated with either mixing intensity
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Fig. 22. Volume-averaged strian rate vs. phase of oscillation. St"1.989 (2 mm).
Fig. 24. Variance of strian rate vs. phase of oscillation. St"0.995 (4 mm).
Fig. 23. Variance of strian rate vs. phase of oscillation. St"0.497 (8 mm).
Fig. 25. Variance of strian rate vs. phase of oscillation. St"1.989 (2 mm).
or power dissipation. In this work, we are particularly interested in establishing such a correlation linking the mean strain rate with the oscillatory Reynolds number, and with the power dissipation in the OBC. Based on the large number of experiments carried out in the OBC, the following correlation, as shown in Fig. 26, was obtained as
Combining Eqs. (7) and (8), the mean strain rate in the OBC can be expressed in terms of e as
c"6;10\ (Re )
(/s) 251)Re )4021,
(7)
which relates the mean strain rate with the oscillatory Reynolds number. The correlation coe$cient is of 98%. In order to relate the mean strain rate with the average energy dissipation per unit mass, e"P/
(8)
c"42.6 e (/s).
(9)
This quantitative correlation is the "rst of its kind in an OBC linking the mean strain rate with the power dissipation. It clearly shows that the mean strain rate is controlled by the power input to the OBC, and using the PIV technique we can quantify the strain rate exerted in the column simply by calculating the power input. 5. Comparison with stirred tank vessels A direct comparison of shear rate between an OBC and a stirred tank (ST) vessel is di$cult due to the nature of the system, the comparison carried out here is merely on an approximate basis. In order to compare the strain rate, we need to choose a common base for the two systems. Metzner and Otto in 1957 proposed that the
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Table 1 The peak values of the volume-averaged strain rates and the variance in the OBC Re x (mm) f (Hz) St (c ) (/s) 4 (p) (/s)
4021 8 1.6 0.497 21.4
3016 8 1.2 0.497 17.5
2011 8 0.8 0.497 11.1
1508 4 1.2 0.995 6.4
1005 8 0.4 0.497 5.1
754 2 1.2 1.989 2.3
503 2 0.8 1.989 1.3
251 2 0.4 1.989 0.7
38.7
30.1
19.5
11.0
8.3
5.0
4.2
1.4
Table 2 A comparison of the mean strain rate in an OBC and STs N (rps)"f (Hz) 0.4 (24 rpm) 0.8 (48 rpm) 1.2 (72 rpm) 1.6 (96 rpm) c (/s) 12 c (/s) - ! c (Oldshue, 12 1983)
Fig. 26. Volume and cycle-averaged strian rate vs. oscillatory Reynolds number.
average shear rate in a ST (c ) is directly proportional to 12 the agitation speed as c "kN, (11) 12 where N is the rotational speed (rpm) and k a constant which depends on impeller design and the type of #uid. Due to its empirical nature there have been a great number of correlation for the k value, e.g. (Calderbank & Moo-Young, 1961) k"B
L \L 4n , 3n#1
(12)
where n is the index of the power-law #uids, in this case, pseudo-plastic #uid and B a system constant that they found to be 11. For a 6-bladed disc turbine, 2 bladed paddles and 3 bladed propeller, Harnby, Edwards & Nienow (1992) found that k"11.5, 10 and 10, respectively. The Metzner}Otto relationship is valid for low Reynolds number #ows in STs, and not for fully turbulent #ows. As our work also belongs to the low Reynolds number #ow regimes, this is in fact the identi"ed common ground for comparison. We decided to take an average value of k"10.5 and apply it to the Metzner} Otto relationship for the same range of revolution per second or oscillation frequency applied. Table 2 shows such a comparison for both systems. In addition, we have
4.2 3.5 No data
8.4 7.1 16.4
12.6 10.1 26.5
16.8 13.3 31.1
also quoted the mean shear data of a 6-in six-blade turbine impeller rotating in a 18-in ST containing water (Oldshue, 1983) for reference. Note that the oscillation amplitude of 8 mm was selected for the comparison, as it was the largest amplitude in our investigation. It is evident that for the same revolution per second the mean strain rate in the OBC is lower than that in a ST vessel estimated using the Metzner}Otto relationship, bear in mind that at such rotational speeds there will be little mixing in a given ST vessel. It is also clear that the mean strain rate in our 50-mm diameter OBC is much smaller than that in a 457-mm diameter ST vessel. This is very encouraging news as the larger the ST vessel, the smaller the strain rate, and the strain rate in our system is more favourable compared with the scale-up ST. This result is particularly important and relevant as OBCs could be applied to biochemical and biomedical "elds involving cell cultures, and a good global mixing complemented with very low shear rate is essential to such processes.
6. Conclusions The PIV has provided quantitative time-resolved information of velocity and strain rate in an OBC. The instantaneous, phase/cycle-averaged properties together with the global characteristics provide a deep insight into the #uid mechanical conditions, the mixing and convective transport phenomena in both axi-symmetric and non-axisymmetric regimes. The quantitative data has opened up a new world of analysis of #uid #ow
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behaviour and transport phenomena in OBCs, which other methods are unable to provide. For the range of the oscillatory Reynolds numbers (251}4021) investigated the volume-averaged mean strain rate varies from 0.5 to 13.5/s, respectively, which is lower than ST vessels. This is particularly important as it suggests that OBCs could be applied to biochemical and biomedical "elds where cultures are involved, and a good global mixing complemented with very low shear rate is essential to such processes. As a prelude to our #occulation study, the results shown in this paper are very encouraging as #oc formation and growth in #occulation processes are sensitive to the magnitude of the strain rate. The potential to simultaneously attain good global mixing and low shears are highly desirable. We have also, for the "rst time, presented the quantitative correlation between the mean strain rate and the power dissipation in an OBC. Using the PIV technique we are able to quantify the strain rate, no longer relying on empirical estimations.
Acknowledgements Authors wish to thank BBSRC for the "nancial support of this project.
References Adrian, R. J. (1991). Particle-imaging techniques for experimental #uid mechanics. Annual Review of Fluid Mechanics, 23, 261}304. Adrian, R. J. (1996). Bibliography of particle velocimetry using imaging methods: 1917}1995. Technical Report 817, USA: University of Illinois. Baird, M. H. I., & Stonestreet, P. (1995). Energy dissipation in oscillatory #ow within a ba%ed tube. Transactions of the Institute of Chemical Engineers, 73, 503}511. Brunold, C. R., Hunns, J. C. B., Mackley, M. R., & Thompson, J. W. (1989). Experimental observations on #ow patterns and energy losses for oscillatory #ows in ducts with sharp edges. Chemical Engineering Science, 44, 1227}1244. Calderbank, P. H., & Moo-Young, M. (1961). Transactions of the Institute of Chemical Engineers, 39, 337}342. Dewhirst, T.P. (1998). Particle-imaging techniques for experimental yuid mechanics. Ph.D. Thesis, Scotland: The University of Edinburgh. Dickens, A. W., Mackley, M. R., & Williams, H. R. (1989). Experimental residence time distribution measurements for unsteady #ow in baf#ed tubes. Chemical Engineering Science, 44, 1471}1479. Elgobashi, S. (1994). On predicting particle-laden turbulent #ows. Applied Science Research, 52, 309}329.
Gao, S., Ni, X., Cumming, R. H., Greated, C. A., & Norman, P. (1998). Experimental investigation of bentonite #occulation in a batch oscillatory ba%ed column. Separation Science and Technology, 33, 2143}2157. Harnby, N., Edwards, M. F., & Nienow, A. W. (1992). Mixing in the process industries (2nd Ed). London: Butterworth-Heinemann Ltd. Hewgill, M. R., Mackley, M. R., Pandit, A. B., & Pannu, S. S. (1993). Enhancement of gas}liquid mass transfer using oscillatory #ow in a ba%ed tube. Chemical Engineering Science, 48, 799}809. Howes, T. (1988). On the dispersion of unsteady yow in ba{ed tubes. Ph.D. thesis, Cambridge University. Howes, T., Mackley, M. R., & Roberts, E. P. L. (1991). The simulation of chaotic mixing and dispersion for periodic #ows in ba%ed channels. Chemical Engineering Science, 46, 1669}1677. Mackay, M. E., Mackley, M. R., & Wang, Y. (1991). Oscillatory #ow within tubes containing wall or central ba%es. Transactions of the Institute of Chemical Engineers, 69(A), 506}513. Mackley, M. R., & Ni, X. (1991). Mixing and dispersion in a ba%ed tube for steady laminar and pulsatile #ow. Chemical Engineering Science, 46, 3139}3151. Mackley, M. R., & Ni, X. (1993). Experimental #uid dispersion measurements in periodic ba%ed tube arrays. Chemical Engineering Science, 48, 3293}3305. Mackley, M. R., Smith, K. B., & Wise, N. P. (1993). The mixing and separation of particle suspensions using oscillatory #ow in ba%ed tubes. Transactions of the Institute of Chemical Engineers, 71(A), 649}656. Mackley, M. R., & Stonestreet, P. (1995). Heat transfer and associated energy dissipation for oscillatory #ow in ba%ed tubes. Chemical Engineering Science, 50, 2211}2224. Mackley, M. R., Tweedle, G. M., & Wyatt, I. D. (1990). Experimental heat transfer measurements for pulsatile #ow in ba%ed tubes. Chemical Engineering Science, 45, 1237}1242. Metzner, A. B., & Otto, R. E. (1957). A.I.Ch.E. Journal, 3, 13}20. Ni, X., Gao, S., Cumming, R. H., & Pritchard, D. W. (1995a). A comparative study of mass transfer in yeast for a batch pulsed ba%ed bioreactor and a stirred tank fermenter. Chemical Engineering Science, 50, 2127}2136. Ni, X., Gao, S., & Pritchard, D. W. (1995b). A study of mass transfer in yeast in a pulsed ba%ed bioreactor. Biotechnology and Bioengineering, 45, 165}175. Ni, X., & Mackley, M. R. (1993). Chemical reaction in batch pulsatile #ow and stirred tank reactors. Chemical Engineering Journal, 52, 107}114. Ni, X., Zhang, Y., & Mustafa, I. (1999). Correction of polymer particle size with droplet size in suspension polymerisation of methylmethacrylate in a batch oscillatory ba%ed reactor. Chemical Engineering Science, 54, 841}850. Ni, X., Zhang, Y., & Mustafa, I. (1998). An investigation of droplet size and size distribution in methylmethacrylate suspensions in a batch oscillatory ba%ed reactor. Chemical Engineering Science, 53, 2903}2919. Oldshue, J. Y. (1983). Genetic engineering news (pp. 46}49). Ra!el, M., Willert, C., & Kompenhans, J. (1998). Particle image velocimetry. Berlin: Springer. Roberts, E. P. L. (1992). Unsteady yow and mixing in ba{ed channels, Ph.D. thesis. Cambridge University.